Multipoles as quantitative order parameters for altermagnetic spin splitting

TL;DR

The paper addresses how altermagnetic spin splitting can be quantitatively linked to local multipoles of charge and magnetization. Using constrained density-functional theory and first-principles calculations on SrCrO3 and LaVO3, it shows that NRSS is not governed solely by the lowest-order multipole but can arise from a superposition of multiple multipole channels, including higher-order terms like octupoles, hexadecapoles, and triakontadipoles. The findings demonstrate that a multi-component order parameter is needed to accurately describe altermagnetism and that NRSS measures based on Brillouin-zone averaging provide robust descriptors across materials and distortions. This framework opens avenues to fingerprint and tailor altermagnetic materials by targeting specific multipole channels and distortion modes.

Abstract

We establish a quantitative relation between the altermagnetic spin-splitting and different higher order multipoles of the charge and magnetization density around the magnetic atoms. Magnetic multipoles such as octupoles or triakontadipoles have been suggested as potential ferroic order parameters for d- and g-wave altermagnetism, respectively, based mainly on qualitative symmetry arguments. We use first-principles-based electronic structure calculations to establish a clear quantitative relation between the strength of the altermagnetic spin splitting and the magnitude of certain local multipoles. We vary the magnitude of these multipoles either by applying an appropriate constraint on the charge density or by varying a corresponding structural distortion mode, using two simple perovskite materials, SrCrO3 and LaVO3, as model systems. Our analysis indicates that in general the altermagnetic spin splitting is not exclusively determined by the lowest order nonzero magnetic multipole, but results from a superposition of contributions from different multipoles with comparable strength, suggesting the need for a multi-component order parameter to describe altermagnetism. We also discuss different measures to quantify the overall spin-splitting of a material, without relying on features that might be specific to only individual bands.

Figures (12)

• Figure 1: Top: Sketch of a charge density exhibiting a nonzero $\mathcal{Q}_{x^2 - y^2}$ quadrupole, characterized by charge accumulation (depletion) along the $x$ ($y$) axis. Bottom: Sketch of a collinear magnetization density with non-zero $\mathcal{O}_{xz}$ octupole. In the red (blue) regions the magnetization points upwards (downwards).
• Figure 2: Dependence of charge and magnetic multipoles corresponding to one of the two Cr atoms in the unit cell on an applied charge quadrupolar perturbation. From left to right, the columns correspond to different perturbations: $s^{202}_{-2}=s_{xy}$, $s^{202}_{-1}=s_{yz}$, $s^{202}_{1}=s_{xz}$, $s^{202}_{2}=s_{x^2-y^2}$.
• Figure 3: Evolution of the different measures for the overall NRSS as function of different induced local charge quadrupoles. The three panels show the global maximum, $\Delta_\text{max}$ (left), the average along the corresponding "ideal" high-symmetry $\mathbf{k}$-paths, $\Delta_\text{avg,hsp}$ (center), and the full BZ average $\Delta_\text{avg,BZ}$ (right).
• Figure 4: Bandstructure of C-AFM SrCrO$_3$ without (left) and with (right) application of a quadrupolar perturbation of $s_{xy}=0.5$ eV. The green circular area highlights a region where problematic bands-crossing occur, leading to potentially incorrect assignment between spin-up and spin-down bands and thus a slight underestimation of the total NRSS.
• Figure 5: Evolution and distribution of the NRSS across band indices in SrCrO$_3$ for increasing quadrupolar perturbation $s_{xy}$. For reference, we indicate all bands that cross the Fermi energy at least at one $\mathbf{k}$-point in the BZ by $\nu_F$ .